Eastham entered the Civilian Pilot Training Program in 1942. He later instructed flying students in El Reno, Oklahoma. Called to active duty with the Air Corps in 1944, he flew the P-51, A-26, and F-6, B-17, and P-47 aircraft. He was one of the original pilots for the Berlin Airlift. He served as a production test pilot for McConnell Aircraft on the F-2H Banshee and F-3h Demon. He was the project test pilot with Hughes Aircraft Company on the Falcon Missile Program at White Sands, New Mexico, flying the F-102 and F-106, and firing record 350 missiles. He was the first pilot to fly the YF-12A and one of the project test pilots on the A-12 and SR-71 aircraft. Eastham was a fellow in the Society of Experimental Test Pilots and a recipient of the United States Air Force Ivan Kincheloe award in 1964.